17‐812
“Production choices with water markets: The role of initial allocations and forward trading”
Philippe Bontems and Céline Nauges
May 2017
Production choices with water markets: The role of initial allocations and forward trading
Philippe Bontems∗ C´eline Nauges†
Abstract
We develop a theoretical model that describes risk-averse farmers’ decisions when facing production risk due to uncertain weather conditions and when irrigation water can be traded on a market. We focus on the role of initial water allocations granted to irrigated farms at the start of the season. The presence of water markets makes the future water price uncertain and hence the value of initial water allocations uncertain.
We analyze the properties of this background risk and study how initial water allocations impact farmers’ land allocation decisions between an irrigated crop and a non-irrigated crop, both characterized by random expected net returns. We then extend the model by permitting irrigation water to be traded ex-ante at a known price (forward market).
Finally, we illustrate our main theoretical findings using simulations. Assumptions made on the distributions of the random variables were chosen to be representative of the situation of irrigated farms located in the Murray-Darling Basin in Australia, where a water market has been in place for several decades.
Key-words: water markets, risk, agriculture.
∗Toulouse School of Economics, INRA & IDEI, University of Toulouse Capitole, France.
†Toulouse School of Economics, INRA, University of Toulouse Capitole, France.
1 Introduction
In regions where water is scarce and rainfall is erratic, irrigated farms face production risk due to the uncertainty of future water availability. To help farmers manage this uncertainty, some countries (e.g., Australia, Chile, the United States) have developed water trading schemes.
Water markets have been shown to increase overall welfare by permitting the reallocation of water between low-value and high-value uses (e.g., Vaux and Howitt, 1984; Garrido, 1998) and to improve efficiency in risk sharing (e.g., Lefebvre, Gangadharan and Thoyer, 2012).
The price of water on the market depends on both demand for, and supply of, irrigation water, and is thus volatile. Price volatility not only adds to the uncertainty of future water availability but also makes the value of farmers’ water rights uncertain. More precisely, the uncertain value of water allocations granted to farmers generates a background risk that is going to impact their production choices if farmers are risk-averse (Eichner and Wagener, 2012). In this article we develop a theoretical model to study how the level of initial water allocations (coupled with the uncertain water price) affects risk-averse farmers’ production decisions.
We consider a risk-averse farmer who receives an initial allocation of water that can be physically used for the production of an irrigated crop or sold on a competitive market. This allocation is regularly revised by water authorities so that, at the beginning of the season, the farmer does not know with certainty the total amount of water that will be available over the entire irrigation season. This uncertainty translates into an uncertain price of water allocations which itself makes the return of the irrigated crop risky. In our setting, the farmer can allocate part of his/her land to a non-irrigated crop, which is also characterized by a random return (due for example - but not restricted - to weather uncertainty). In this model, a farmer’s decision about the allocation of land between two risky crops can be seen as a portfolio decision of a risk-averse agent owning an initial (uncertain) wealth and having the possibility of investing in two (possibly correlated) risky assets, which is a setting similar
to Eichner and Wagener (2012, hereafter EW).1
We follow EW in assuming that farmers’ preferences are characterized by a utility function defined over the mean and variance of their random wealth. Under some assumptions about the form of farmers’ risk preferences we show, in the same vein as EW, how the level of initial allocations impacts farmers’ allocation of land between the irrigated crop and the non- irrigated crop. Next, we extend EW’s analysis by studying the implementation of a forward market for water allocations. The possibility of trading water forward at a certain price offers farmers a new option for managing the risk brought about by the uncertainty about the price of water and hence can modify farmers’ land allocation decisions.
Finally, we run simulations to illustrate our main analytical findings. Distributions of the random variables and levels of the parameters of interest were chosen based on conditions observed in the Murray-Darling Basin (MDB) in Australia. The MDB, which covers about 14% of Australia’s land mass, is the main source of irrigated agricultural production in the country and hosts one of the most active water markets in the world. The research question, the underlying assumptions about the sources of risk, and the timing of farmers’ decisions were primarily motivated by the functioning of the market for water allocations in the MDB.
Water allocations can be seen as temporary water rights. They correspond to the actual volume (or allocation) of water assigned to the farmer on the basis of his water entitlements (or permanent water rights). Water allocations are announced at the beginning of the season and revised regularly based on water storage levels and expected inflows. The trading of water allocations is used as a short-term risk management tool by farmers in the MDB and, in some years, more than 30% of announced water allocations are traded (Zuo, Nauges and Wheeler, 2015). The level of the water allocations (expressed as a percentage of the nominal quantity of water entitlement) and their price can vary substantially across years and within a season, especially in times of drought or when storage levels are low. A forward market for
1The impact of a background risk in wealth on financial decisions such as the portfolio problem has received considerable attention in the finance and insurance literature, both in the expected utility framework and in the mean-variance context. See e.g. Tsetlin and Winkler (2005), Jiang, Ma and An (2010), Li (2011) and Eeckhoudt and Gollier (2013) for a survey.
water allocations was launched in the MDB in 2014.
Section 2 describes how our article relates to and contributes to the existing literature.
Section 3 describes the farmer’s risk preferences and the theoretical decision model where the main choice variable is land allocation between the irrigated and non-irrigated crops. In Section 4, we focus on the role of initial allocations granted to the farmer at the beginning of the season. We first discuss the characteristics of the background risk induced by these initial water allocations and the uncertainty of future water prices. Second, we show how initial allocations impact the farmer’s optimal land use decision. Section 5 extends the theoretical model to allow for forward trading at the start of the season. Section 6 provides an illustration of our main theoretical findings using simulations, and Section 7 concludes.
2 Related literature
The literature describing the behavior of risk-averse firms in the presence of uncertainty dates back to the 1970s and the work of Baron (1970) and Sandmo (1971). Our article relates primarily to studies that have investigated the production decisions of risk-averse agents in the presence of uncertainty and regulatory instruments (market-based policies).
In this literature however, the main focus has been on pollution quotas or emissions taxes and the impact of uncertainty and risk aversion on firms’ polluting emissions and compliance levels (e.g., Ben David et al., 2000; Baldursson and von der Ferh, 2004, 2012).2 One exception for the agricultural sector is Babcock (1990) who studied the production decisions of farmers facing marketing quotas.
Our article also relates to the literature on the design of quota systems in general, and the dependence of market outcomes on initial allocations in the presence of uncertainty and risk aversion, in particular. This dependence was proved in Baldursson and von der Ferh (2004) in a theoretical setting. The authors showed that risk-averse firms with small initial
2These articles differ in terms of the source of the uncertainty faced by the agents: pollution permit prices in Ben David et al. (2000); total number of polluting firms or firm’s level of pollution in Baldursson and von der Ferh (2004); and input and/or output prices in Baldursson and von der Ferh (2012).
quota holdings are expected to overinvest in abatement technology, while those with large initial quota holdings are expected to underinvest.3
Other related articles include Hahn and Stavins (2011) which listed six conditions under which the final allocation of permits (after trade has occurred) and the permit price are not independent of the initial allocation of permits,4and Fowlie and Perloff (2013) who empirically tested the independence property using data from the Southern California’s Regional Clean Air Incentives Market (RECLAIM).
As far as we know, the role of initial allocations in the context of water markets has not been addressed in the literature. Recent analyses of water markets have tended to study farmers’ decisions in terms of water trading using different modeling approaches (see, for example, Ranjan and Shogren, 2006, using a non-expected utility framework and Cristi, 2007, using a consumption-based asset pricing model). Calatrava and Garrido (2005) studied how the variability in profits (caused by the uncertainty of water allocated to the farm) is modified when a market for water is put in place, and showed that risk exposure is reduced for market users, whether they participate as sellers or buyers. In Lefebvre, Gangadharan and Thoyer (2012) data from laboratory experiments were used to study the impact of differentiated water rights (high and low priority rights, the former being served first when water is scarce) on cost effectiveness and risk management properties of water markets in the presence of transaction costs.
Finally, our article adds to the sparse literature investigating the production decisions of non-risk neutral farmers in the presence of multiple risks. Among the few articles that studied two or more risks are Coyle (1999), Isik (2002), and Serra et al. (2006). The present article is closer in spirit to Coyle (1999) and Serra et al. (2006) who analyze farmers’ production
3There is an assumption of zero transaction costs in Baldursson and von der Ferh (2004). Montero (1997), in a setting with transaction costs and uncertainty, showed that initial allocations have an impact on market outcomes (equilibrium prices) and welfare even in the case of risk-neutral polluting firms which receive an initial allocation of pollution permits that can be traded.
4The six conditions are: 1) transaction costs (such as those due to search and information or bargaining); 2) market power; 3) uncertainty about the future price of permits (in the presence of risk aversion and transaction costs); 4) conditional allocations of permits (e.g., conditional on output in the previous period); 5) non-cost minimizing behavior (e.g., for public firms); and 6) differentiated regulations.
decisions in a mean-variance framework, while allowing for non-neutral risk preferences.
3 The farmer’s decision model
3.1 A portfolio problem
We consider a farmer who has the possibility of allocating his land of size l to either an irrigated crop or a non-irrigated crop. We denote the share of land devoted to the irrigated production byx and the share assigned to the non-irrigated production by 1−x. Irrigation water can be bought or sold on a competitive market at pricer. From an ex-ante perspective, the water priceris random with mean ¯rbecause of the uncertainty about the total amount of irrigation water that will be available over the season. The (per hectare) net benefit of both the irrigated crop and the non-irrigated crop are random and denoted R and εrespectively, with means ¯R and ¯ε.
Randomness in the return of the irrigated crop is mainly driven by the uncertain water price while randomness in the return of the non-irrigated crop might be driven by variability in weather conditions through its impact on crop yield and/or by output price variability.
Finally, we denote e0 ≥0 as the initial level of water allocations which is made available to the farmer at the beginning of the growing season. The value of this initial endowment is unknown ex-ante due to the uncertainty of the water price.
The timing of farmers’ decisions is the following: at the start of the growing season (spring, in the context of the MDB in Australia), initial water allocations e0 granted to the farmers are announced and the farmers decide on the land area that will be planted with the irrigated crop (x). At that time, the quantity of irrigation water that will be available in the summer (when the irrigated crop is in greatest need of water) and the price at which water will be sold, are not known with certainty. The decision made in the spring about the share of land to be irrigated (x) can thus be seen as an ex-ante decision, before the uncertainty is resolved.
In the summer, weather conditions have been observed and information on water availability is fully revealed, so farmers no longer face any uncertainty. They have full information when
deciding on the quantity of water used to produce the irrigated crop and on the quantity of water traded on the market (ex-post decisions). In the following we focus only on farmers’
optimal (ex-ante) decisions in terms of land allocation (x).
The farmer’s ex-post (final) wealth, which corresponds to the sum of the net returns of the irrigated and non-irrigated productions, plus his/her initial (certain) wealthw0, and the value of the initial water allocations e0, is written as follows:5
w=w0+xlR+ (1−x)lε+re0.
Sincewis linear inl, we can normalizelto 1 without any loss of generality (it is sufficient to redefine e0 as the endowment per hectare). Hence, ex-post wealth is equivalently written:
w=w0+xR+ (1−x)ε+re0.
The decision problem of the farmer, i.e. how to allocate land between two risky assets with uncertainty about its initial wealth, is akin to a standard portfolio choice problem. The possibility of trading water on a market allows ex-post flexibility for the producer but a strictly positive initial endowment also generates a cost in terms of risk exposure through the introduction of an (additive) background risk potentially correlated with the two risky activities,R and ε.
Since water is used as an input in the production of the irrigated crop, we expect a higher water price r to induce a lower net benefit R for the irrigated production. Also, because higher water prices are more likely to be observed in dry conditions, we expect a negative correlation betweenrandε, the net benefit of the non-irrigated crop. Hence we assume in the following that the water price r is negatively correlated with bothR and ε, with covariances σrR<0 andσrε<0, respectively.
5Implicitly, the ex-post net returnR of the irrigated crop can be defined as: R = maxz(pf(z, θ)−rz)x withz, the quantity of water supplied through irrigation and sold on a competitive market at price r; f, a production function that is assumed to be increasing and concave inz;θ, a random production shock, andp the crop price.
3.2 Assumptions on farmers’ preferences
Following EW, we assume that preferences over the random ex-post wealth can be represented by a two-parameter utility function V(µ, v) where µ and v are the expected value and the variance of the ex-post wealth, respectively. The expected ex-post wealth µis as follows:
µ(x, e0) =w0+xR¯+ (1−x)¯ε+ ¯re0.
By denoting Y as the final wealth in the absence of any water endowment, Y = w0 + xR+ (1−x)ε, we rewrite µ(x, e0) = ¯Y(x) + ¯re0 with ¯Y(x) =w0+xR¯+ (1−x)¯ε. We assume in the following that the average return of the irrigated crop is larger than the average return of the non-irrigated crop: ¯R >ε.¯6
The variancev of the ex-post wealth is written as:
v(x, e0) =x2vR+ (1−x)2vε+e20vr+ 2x(1−x)σεR+ 2xe0σrR+ 2(1−x)e0σrε, wherevi is the variance ofifori=R, ε, r. Denoting the covarianceσrY(x) =xσrR+ (1− x)σrε (which is <0 given our assumptions about correlations), and denoting the variance of Y asvY(x) =x2vR+ (1−x)2vε+ 2x(1−x)σεR, we obtain:
v(x, e0) =vY(x) +e20vr+ 2e0σrY(x).
It is easy to check that the variance is a quadratic convex function ofxas ∂∂x2v2 =vY00(x) = 2vR−ε>0 where vR−ε is the variance ofR−ε.7
Let us now turn to assumptions about farmers’ risk preferences. First, we assume thatV is aC3 function with
Vµ>0,Vv <0,Vµµ <0,Vvµ >0.
The fact that Vv < 0 and Vµµ < 0 reflects that preferences are characterized by risk aversion (while risk neutrality would correspond to Vv = 0).8 We denote the marginal rate
6This condition implies that the expected wealthµis increasing in the land sharexdevoted to the irrigated crop.
7For now we leave the sign ofσ0rY(x) =σrR−σrεunspecified but, in the following, most of the time we use the assumption that the water price r is more (negatively) correlated with R than withε: σ0rY(x) = σrR−σrε<0.
8The assumptionVvµ>0 is the analogue of prudence in the expected utility framework.
of substitution between µand v by
α(µ, v) =−Vv
Vµ >0
which is the mean-variance counterpart of the Arrow-Pratt measure of absolute risk aversion.
We assume both decreasing absolute risk aversion (DARA) preferences (αµ<0) and variance vulnerability (αv > 0). Under DARA preferences the marginal willingness to pay for a reduction in variance decreases when expected wealth raises. Variance vulnerability means here that the marginal willingness to pay for a reduction in variance,α, is increasing as final wealth becomes more volatile. When these two conditions are met, the agent is said to be risk vulnerable (Eichner, 2008).
Finally, to ensure second-order conditions in optimization programs, we consider that V(µ, v) is quasi-concave or equivalently that:
ααµ+αv >0 as demonstrated by Eichner (2008).9
3.3 Farmers’ optimal land use decision
The farmer chooses the land allocation x such as to maximize his (mean-variance) utility of final wealth:
maxx U(x, e0)≡V(µ(x, e0), v(x, e0)).
For an interior solutionx∗∈(0,1), the first-order condition is written:
∂U(x∗, e0)
∂x =Vµ
∂µ
∂x +Vv
∂v
∂x = 0 or equivalently
∂µ
∂x/∂v
∂x =α(µ, v). (1)
Condition (1) indicates that the portfolio frontier (or transformation curve) with slope
∂µ
∂x/∂v∂xmust be tangent to an indifference curve with slopeαat the optimum. In what follows,
9As shown by Lajeri-Chaherli (2002), this condition together with DARA preferences (αµ<0) is equivalent to the concept ofproper risk aversion, which is more demanding than risk vulnerability.
we disregard the analysis of corner solutions and we assume that the second-order condition
∂2U(x∗,e0)
∂x2 ≤0 is satisfied.10
Since ∂µ∂x = ¯R−ε >¯ 0, it follows from (1) that necessarily ∂x∂v >0 at the optimum. Hence, (1) can be rewritten as:
∂µ
∂x =α∂v
∂x. (3)
At the optimum, the marginal benefit of x, which is measured by the marginal impact of x on the mean wealth (∂µ∂x > 0), is equal to its marginal cost, which here corresponds to the marginal increase in variance of wealth (∂v∂x) times the marginal willingness to pay for a reduction in variance (α).
This result confirms that, in the presence of risk aversion, the initial allocatione0 impacts the ex-ante choicex∗, which is in line with Hahn and Stavins (2011)’s result that the property of the outcome being independent of the initial endowment does not hold in the presence of risk aversion.11
4 The impact of initial water allocations
Providing an initial endowment of watere0 to the farmer introduces a background risk that impacts his/her ex-ante decisions. We first describe the characteristics of this background risk and then we show how these allocations impact the farmer’s optimal land use decision.
4.1 Fairness and desirability of the background risk
Contrary to EW, the background risk in our model is always fair as
∂µ
∂e0 = ¯r >0,
10We actually prove in Appendix A that
∂2U(x∗, e0)
∂x2 ≤0⇔(ααµ+αv)
∂v
∂x 2
+α∂2v
∂x2 ≥0. (2)
Given the convexity of the variance with respect tox, the quasi-concavity ofV(µ, v) is a sufficient condition for (2) to hold.
11A risk-neutral producer (Vv= 0⇒α= 0) would choosex∗= 1 if ¯R >ε¯andx∗= 0 if ¯R <ε. If ¯¯ R= ¯ε, the farmer is indifferent between allocating his land to the irrigated crop and the non-irrigated crop.
i.e., a larger initial water allocation e0 always raises the expected ex-post wealth.12
The background risk is actually ariskonly when its upscaling never decreases the variance of final wealth:
∂v
∂e0 = 2e0vr+ 2σrY(x)≥0,
which happens if and only if the initial endowment is large enough for a given x:
e0 ≥e˜0(x)≡ −σrY(x)
vr . (4)
When condition (4) holds, we can equivalently say that the water endowment e0 is risk increasing. If the opposite condition holds, the initial water endowment is said to be risk decreasing.13
The background risk will be said to be undesirable if its marginal impact on utility is negative. The marginal effect of the initial endowment e0 on utility is:
∂U(x, e0)
∂e0
= Vµ
∂µ
∂e0
+Vv
∂v
∂e0
= Vµ
∂µ
∂e0 −α ∂v
∂e0
which holds for anyxincludingx∗due to the envelope theorem. The background risk created by the water endowment is thus undesirable if and only if
∂µ
∂e0
≤α ∂v
∂e0
, (5)
which allows the following proposition, which holds for any xincludingx∗, to be established:
Proposition 1 The fair background risk created by the water endowment e0 is undesirable if and only if it is sufficiently risk increasing (∂e∂v
0 ≥ αr¯ >0).
We are able to show that, at the optimum x=x∗(e0), low values of e0 are risk reducing and large values are risk increasing, with ambiguity in between (see Appendix B).
12EW focus on unfair background risks, as does most of the literature.
13Note that assumingσ0rY(x) =σrR−σrε<0 implies that the threshold level ˜e0(x) is larger whenxrises.
In other words, a larger land share devoted to the irrigated crop implies that the water endowment is more often risk decreasing.
4.2 Impact of the initial allocations on farmers’ land use choices
Following EW, differentiating (1) with respect to e0 and using the second-order condition allows us to establish that:
∂x∗
∂e0
=s
−αµ∂µ
∂e0
| {z }
ME
+
−αv ∂v
∂e0
| {z }
VE
+ −α
∂v
∂x
∂2v
∂x∂e0
! .
| {z }
CE
(6)
As shown in (6), the marginal impact of e0 on the optimal allocation of land can be decomposed into three terms: a mean effect (ME); a variance effect (VE); and a correlation effect (CE). These are discussed in turn.
1. The mean effect (ME): it is always positive under DARA preferences (αµ<0) since the background risk is fair (∂e∂µ
0 = ¯r >0). Intuitively, increasinge0 implies a larger positive background wealth and hence, under DARA preferences, it is optimal for the farmer to take more risk by allocating more land to the (riskier) irrigated crop.
2. The variance effect (VE): under variance vulnerability (αv >0), the sign of the variance effect is positive when e0 is risk decreasing and negative whene0 is risk increasing.
3. The correlation effect (CE): at the optimumx∗ is risk increasing (∂v∂x >0) and we have
∂2v
∂x∂e0 = 2σ0rY(x∗) = 2 (σrR−σrε),
σrR<0 andσrε<0. Hence, when the water price is more negatively correlated withR than with ε(σrR−σrε <0), the correlation effect is positive and pushes x∗ up.14 The intuition is the following: an increase in e0 raises the background risk which induces the risk-averse agent to diversify his portfolio by investing more in the asset that is more negatively correlated with the background risk, in our case the irrigated crop (if we assume σrR< σrε<0).
14Conversely, if the water price were more negatively correlated with the non-irrigated crop, then the correlation effect would be negative. It would be zero ifσrR=σrε.
When e0 is risk decreasing at the optimum, the overall effect is positive since all three components push x∗ up. When e0 is risk increasing, the variance effect is negative and the sign of the combined effect is ambiguous.15
These findings are summarized in the following proposition.
Proposition 2 Under the assumption that the water price is more negatively correlated with the return of the irrigated crop, an increase in the background risk (induced by a larger water endowment) induces the farmer to increase the share of irrigated land (x):
(i) when the water endowment is risk decreasing at the optimum; or
(ii) when the water endowment is risk increasing at the optimum, if and only if the correlation effect and the mean effect are strong enough to outweigh the variance effect (i.e. CE+ M E >−V E >0).
5 Introducing the possibility of forward trading
In this section we investigate how a farmer’s land allocation decision is impacted if the farmer has the possibility of buying or selling all, or a portion of, his initial water endowment on a forward market. The theoretical model described in the preceding section is extended and includes two ex-ante decisions, i.e. decisions made by the farmer at the start of the growing season before the uncertainty is resolved: (1) how much land to allocate to the irrigated crop (xf) and (2) the quantity of water to be traded (either bought or sold) on the forward market (y).
15Reinterpreting Proposition 4 in EW, it is possible to show that whene0is undesirable and when preferences exhibit properness, then the negative variance effect always outweighs the positive mean effect. Indeed, multiplying both sides of (5) by −αµ > 0 (DARA preferences) and adding −αv∂v
∂e0 to both sides of the inequality, we obtain:
−αµ
∂µ
∂e0
−αv
∂v
∂e0
≤ −(αµα+αv)∂v
∂e0
≤0.
The last component of the inequality follows from the quasi-concavity ofV (αµα+αv>0) and from ∂e∂v
0 >0.
5.1 Farmers’ ex-post wealth
With forward trading, the farmer can trade a quantity of water ex-ante at a known and fixed market price rf, so his ex-post (final) wealth is:
wf =w0+xfR+ (1−xf)ε−rf(y−e0) +ry, (7) where y is the post-trade water endowment. The initial endowment e0 now only generates a non-random wealth effect, its value being the certain price rf, except of course when the farmer decides not to intervene on the forward market (y =e0). The farmer can manage his exposure to the risk r through his choice of both xf and y (the farmer can choose to be a buyer if y > e0 or a seller if y < e0).
We denote µf the expectation of final wealth under forward trading, which is written:
µf(xf, y) = w0+rfe0+xfR¯+ (1−xf)¯ε+y(¯r−rf)
= µ(xf, y) +rf(e0−y).
The expected wealth µf with forward trading is equal to the expected wealth µ without forward trading wheree0 is replaced byy,plus an additional termrf(e0−y) that is positive if the producer is a seller and negative if he/she is a buyer.
The variance of the final wealthvf is written as the variance v without forward trading except that the initial endowment e0 is being replaced by the post-trade endowmenty:
vf(xf, y) = xf2vR+ (1−xf)2vε+y2vr+ 2xf(1−xf)σεR+ 2xfyσrR+ 2(1−xf)yσrε
= v(xf, y).
Note that the introduction of the forward market enlarges the decision set of the farmer.
As a consequence, the farmer’s utility is weakly increasing at the optimum compared to the situation without forward trading and the farmer can always reach his/her former utility level by not intervening on the forward market.
5.2 A farmer’s optimal decisions on land use and forward trading
With forward trading, the problem for a risk-averse farmer is to maximize his/her utility U(xf, y;e0) ≡ V(µf(xf, y), vf(xf, y)) over two decision parameters: the share of land allo- cated to the irrigated crop (xf); and the quantity of water traded ex-ante on the forward market (y). Whenever it exists, an interior solution (x∗f, y∗) is characterized by the two following first-order conditions:
∂µf
∂xf
∂vf
∂xf
=α(µf, vf) =
∂µf
∂y
∂vf
∂y
. (8)
Second-order conditions hold under proper risk aversion and provided correlation is not too strong as shown in Appendix C.16
As in the situation without forward trading, the optimal choice of xf is such that the farmer faces a trade-off between an increase in expected wealth (∂µ∂xf
f = ¯R−ε >¯ 0) and an increase in the variance of wealth (∂v∂xf
f > 0). Similarly for the optimal choice in terms of trading on the forward market y∗, ∂µ∂yf = ¯r−rf and ∂v∂yf have the same sign but this sign depends on whether ¯r−rf is positive or negative.
When anticipations are biased upwards (rf >¯r), the farmer chooses y to equate his/her marginal benefit in terms of a reduction in the variance of wealth with its marginal cost in terms of expected wealth losses. Conversely, when anticipations are biased downwards, an increase in y leads to a marginal benefit in terms of expected wealth and a marginal cost in terms of increased variance. Lastly, when anticipations are not biased (rf = ¯r), the post- trade water endowmenty∗is chosen such that the variance of wealth is minimized, i.e.∂v∂yf = 0.
In what follows, we discuss conditions under which the farmer is a net buyer or a net seller on the forward market. If we assume that U(xf, y) is concave in (xf, y), then it is possible
16Under risk neutrality (Vv= 0), only expected wealth matters and we straightforwardly obtain that when anticipations are pessimistic with respect to ¯r, i.e. biased upwards (rf >r), it is optimal to sell¯ e0 ex-ante.
On the contrary, when anticipations are optimistic, i.e. biased downwards (rf <¯r), it is optimal to buy as much water as possible ex-ante and to re-sell it ex-post. And when anticipations are unbiased (rf = ¯r) forward trading is of no interest andy∗is indeterminate.
to derive a simple condition upon which it is optimal to sell or buy on the forward market.
Consider the first-order derivative with respect to y and evaluated aty=e0: dU
dy y=e0
=s ∂µf
∂y y=e0
−
α(µf, vf)∂vf
∂y
y=e0
dU dy y=e0
= (¯s r−rf)−α∗∂v(xf∗, e0)
∂e0
where α∗ ≡α(µ(xf∗, e0), v(xf∗, e0)) because xf∗ is optimal ify=e0. Given concavity, if dUdy
y=e0
>0, theny∗> e0 necessarily and the producer will be a buyer on the forward market. Conversely, if dUdy
y=e0
<0, then y∗ < e0 and the producer will be a seller.
Proposition 3 If rf < r¯−α∗∂v(x∂ef∗,e0)
0 then the producer is a net buyer on the forward market. If rf >r¯−α∗∂v(x∂ef∗,e0)
0 then the producer is a net seller on the forward market. If rf = ¯r−α∗∂v(x∂ef∗,e0)
0 then the producer is not active on the forward market.
Intuitively, under risk neutrality, the position on the forward market, whether buyer or seller, only depends on whether ¯r−rf is positive or negative. By contrast, under risk aversion, being a buyer or a seller also affects the variance of final wealth. Locally aty=e0, this effect depends on whether the water endowment is risk increasing or risk decreasing. In the case of risk increasing allocations (∂e∂v
0 > 0), risk aversion gives incentives to become a seller on the forward market. It follows that it is only if anticipations are sufficiently pessimistic, i.e.
biased downward so that rf <r¯−α∗∂e∂v
0, that it is optimal to act as a buyer when e0 is risk increasing. With unbiased expectations, a risk-averse producer will be a seller when ∂e∂v
0 >0.
Conversely, when e0 is risk decreasing, then risk aversion more often makes the producer a buyer. With unbiased expectations, a risk-averse farmer will be a buyer when ∂e∂v
0 <0.
5.3 The impact of initial water allocations
We now turn to the comparative statics of the solution (x∗f, y∗) with respect to the initial water endowment e0. First, note that the possibility of forward trading makes the initial
allocation of water e0 always desirable because its value is no longer random.17
Deriving the system of first-order conditions (8) and rearranging, we obtain (cf. Appendix D):
dx∗f de0
=s Uye0Uxy −Uxe0Uyy s
= (¯r−rf) (σrε−σrR)
| {z }
indirect correlation effect
+ R¯−ε¯ vr
| {z }
direct mean effect
. (9)
The marginal impact of e0 on decision x∗f can be decomposed into a direct mean effect and an indirect correlation effect, which allows the following proposition to be derived.
Proposition 4 An increase in the initial endowment e0 always induces the farmer to take more risk in land use if and only if anticipations are either unbiased or optimistic or not too pessimistic, i.e. rf ≤r¯+(σ( ¯R−¯ε)vr
rε−σrR).
To understand this result intuitively, first consider that anticipations are unbiased. When
¯
r =rf,y∗ only aims at minimizing the variance and consequently the initial water allocation whose value is non-random does not impact the marginal utility of y (Uye0 = 0). Hence, only the positive direct effect −Uxe0Uyy
=s R¯−ε¯
vr remains in (9) and is explained by the fact that raising e0 also raises the marginal utility of xf (Uxe0
= ¯s R−ε >¯ 0).Intuitively, a farmer with a larger e0 gets richer and, with DARA preferences, will find it optimal to take more risk. This positive direct effect is similar to the mean effect identified in the absence of forward trading (see Proposition 2).
Regarding the indirect effect (¯r−rf) (σrε−σrR), the intuition is the following. Suppose that ¯r > rf, then being richer (through an increase in e0) stimulates the investment in y that leads to a positive expected return, but at the expense of raising the background risk.
Concerning the correlation effect in Proposition 2, it is optimal to diversify the portfolio by investing more in the asset that is more strongly negatively correlated with r, i.e. in xf becauseσrε−σrR>0. Conversely, when ¯r < rf, the indirect effect is negative since a positive
17Using the envelope theorem, dU de0
= ∂V
∂µf
∂µf
∂e0
= ∂V
∂µf
rf >0,∀e0.
expected return can be obtained by selling some of the initial allocation, thereby reducing the size of the background risk. It is only if anticipations are sufficiently pessimistic that the indirect effect outweighs the positive direct effect.
Similarly, in Appendix D, we prove that:
dy∗ de0
=s Uxe0Uxy−Uye0Uxx=s R¯−ε¯
(σrε−σrR) + (¯r−rf)vR−ε.
Hence, a similar result to Proposition 4 holds fory∗, except that the condition is now:
rf ≤r¯+( ¯R−ε) (σ¯ rε−σrR) vR−ε
.
The sign of the direct mean effect (¯r−rf)vR−ε depends on whether anticipations make it worthwhile to invest or to disinvest in the water allocation. The indirect correlation effect
R¯−ε¯
(σrε−σrR) is positive indicating that being richer due to an increase ine0pushes xf up and, by complementarity because of the correlation effect σrε−σrR>0, gives incentives to increase y.
Finally, using the first-order conditions (8), we are also able to show that the ex-ante optimal production choicex∗f and the post-trade water endowmenty∗ are linearly dependent (cf. Appendix E). This linear dependence contrasts with the situation without forward trading where the relationship between land use and water endowment was typically non- linear.
5.4 Comparison of optimal land use with and without forward trading
We discuss whether the introduction of forward trading induces the farmer to take more risk in terms of land use compared to the situation without forward trading. This could be the case because the essence of forward trading is to manage the risk brought about by the initial water endowment e0. Formally, we would like to compare x∗(e0) ∈ arg maxxV(µ(x, e0), v(x, e0)) with x∗f(e0) ∈arg maxxfmaxyV(µ(xf, y) +rf(e0−y), v(xf, y)). Unfortunately, there is no general result in that perspective and the comparison remains an empirical issue. Nevertheless
we can express the comparison result as the outcome of two potentially conflicting forces as made clear below.
From (7), let us rewrite the final wealth as follows:
wf =w00+xfR+ (1−xf)ε+ry
which is similar to the final wealth without forward trading, except that y replaces e0 (size of the background risk) and that the non-random wealth is now w00=w0+rf(e0−y). For a given post-trade allocation y, there are thus the two possibilities described below.
Consider first that the producer is a net buyer (y > e0) on the forward market. Compared with the case without forward trading, he/she optimally chooses a lower non-random wealth w00 < w0 in exchange for a larger size for the background risk. Under DARA preferences, we know that a lower non-random wealth would induce the producer to take less risk (wealth effect). But under the conditions of Proposition 2 and assuming that the variance effect is dominated by the mean and the correlation effects, a larger background risk would induce the producer to take more risk, i.e. to increase the share of land allocated to the irrigated crop. There is thus some ambiguity with respect to the impact of the buyer’s strategy on the optimal choice with respect toxf depending on the relative magnitude of the wealth and background risk effects.
Actually a similar trade-off exists when the producer is a net seller (y < e0). He/she optimally chooses a higher non-random wealth w00 > w0 in exchange for a lower size for the background risk. Once again, there is ambiguity with respect to the impact on the optimal choice for xf because, on the one hand, under DARA, a higher non-random wealth induces the producer to take more risk while, on the other hand, according to Proposition 2, a lower background risk may entail the producer to take less risk. To sum up, we have the following proposition.
Proposition 5 If the farmer is a seller and if the background risk effect induces the farmer to take more risk, then the forward market pushes xf up. If the farmer is a buyer and
if the background risk effect induces the farmer to take less risk, then the forward market diminishes xf. Otherwise, there is ambiguity and the outcome depends on the sign and the relative magnitude of the wealth effect and the background risk effect.
If we assume further thatU(xf, y;e0) is concave in (xf, y), then we can prove that when- ever the forward market allows the marginal cost of xf expressed as α∂x∂v
f to be reduced, then the farmer takes more risk with forward trading. To see this, denote ˆy(xf) the opti- mal post-trade endowment for a given land use xf. It is given implicitly by the first-order condition:
∂µf(xf,y(xˆ f))
∂y =α(µf(xf,y(xˆ f)), vf(xf,y(xˆ f)))∂v(xf,y(xˆ f))
∂y . (10)
For the sake of notation, we denote ˆα∗ the value of α(µf(xf,yˆf(xf)), v(xf,y(xˆ f))) when xf
is set equal tox∗(e0), the optimal share of irrigated land in the absence of forward trading.
Computing the first-order condition with respect toxf and evaluating it atxf =x∗(e0), we obtain:
dU dxf
xf=x∗(e0)
=s ∂µf
∂xf
xf=x∗(e0)
−αˆ∗∂v(x∗(e0),y(xˆ ∗(e0)))
∂xf . (11)
As ∂µ∂xf = ∂µ∂x = ¯R −ε¯ whatever x and making use of the first-order condition (3) in the absence of forward trading, we can rewrite (11) as the difference between the marginal cost of land usex without and with forward trading:
dU dxf xf=x∗
=s α∗∂v(x∗(e0), e0)
∂xf −αˆ∗∂v(x∗(e0),yˆf(x∗(e0)))
∂xf It follows that, by concavity, if dxdU
f
xf=x∗(e0)>(<)0, then this implies thatx∗f >(<)x∗. We conclude that the ability of forward trading to push x up depends on whether water trading is able to decrease the marginal cost of x.18
18The ambiguity remains even in the case of unbiased expectations (¯r =rf). In that situation, we have α∗>αˆ∗becauseµ(x∗, e0) =µf(xf∗
, y∗),v(x∗, e0)< v(xf∗
, y∗) asy∗minimizes the variance in that case and becauseαv>0. We also have that∂2v/∂x∂y= 2(σrR−σrε)<0. Hence, it is only if ˆy(x∗(e0))> e0 that we can conclude thatx∗f > x∗.
6 Illustrative simulations: the case of the MDB in Australia
In this section we use simulated data to quantify the impact of initial water allocations on a farmer’s land allocation decisions between an irrigated crop and a non-irrigated crop under conditions experienced in the MDB in Australia. We compare land allocation decisions with and without forward trading, and under both normal and dry conditions. Prices of water allocations are higher and yields and net benefits are lower in dry years compared to average (or normal) years.19
6.1 Choice of distributions and parameters
We chose cotton as the irrigated crop and dry-land wheat as the non-irrigated crop under normal conditions but irrigated wheat and dry-land wheat under dry conditions because water-intensive crops such as cotton and rice are not usually grown in dry years.20 Our model includes three non-independent random variables: r, the water price; R, the net benefit per hectare for the irrigated crop; and ε, the net benefit per hectare for the non- irrigated crop. The distribution of the price of water allocations was calibrated using water trading data produced by the Australian National Water Commission (NWC). The NWC database contains records of thousands of individual water trades (volume, price and date of the transaction) in the MDB, from August 2007 until June 2010. From these data, an average water price could be calculated weekly (weights equal to the volume traded in each transaction were used to calculate the average). We calibrated the distribution of water prices using the entire series for the normal conditions scenario since it encompasses episodes of both lower-than-average and higher-than-average rainfall. For the dry conditions scenario, calibration was based on the first half of the period from August 2007 to June 2009 which was characterized by sustained lower-than-average rainfall. Data analysis and statistical tests
19At the start of the season farmers are informed about water storage in dams and the initial water allocation also provides an indication of whether dry or normal conditions are expected over the season. So it is reasonable to assume that farmers, when making their land allocation decisions, know whether a dry or a normal season is more likely to prevail.
20When a drought prevails, the price of water allocations is high and cotton farmers are usually better off not producing and selling their water allocations instead.
Table 1: Parameters for the distribution ofr,R and ε
Variable Dry conditions Normal conditions
r Lognormal (mean = 5.91; std dev = 0.42) Lognormal (mean = 5.57; std dev = 0.61) R Beta (2; 3.3) with bounds (-1,000; 1,200) Beta (1; 1.3) with bounds (0; 4,500) ε Beta (2; 3.3) with bounds (-500; 240) Beta (2; 3.3) with bounds (70; 1,200)
r: price of water allocations (AUD/ML).
R: net return of the irrigated crop (AUD/ha).
ε: net return of the non-irrigated crop (AUD/ha).
showed that a two-parameter log-normal distribution provided a good fit of the water price series.
Beta distributions are used to describe net returns of the irrigated and non-irrigated crops. Shape parameters, lower and upper bounds of each of the four beta distributions (cotton and dry-land wheat in normal conditions and irrigated wheat and dry-land wheat in dry conditions) were determined based on reports on MDB farms’ performance published by the Australian Bureau of Agricultural and Resource Economics (ABARE) for the dry conditions scenario and the New South Wales Department of Primary Industries for the normal conditions scenario.21 Table 1 provides a summary of the distributions and parameters chosen for the two scenarios.
The three random variables are not independent: R depends onr (sincer is the price of one essential input) and we expect these two random variables to be negatively correlated.
We set the Spearman rank correlation coefficient at ρ1(r, R) = −0.3. If uncertainty in the non-irrigated activity is mainly driven by rainfall patterns, we expect the net benefit per
21Beta and log-normal distributions are commonly used to describe distributions of crop yields, which are often assumed to be skewed (Babcock and Hennessy, 1996). However, there is not much discussion in the literature on the distribution of agricultural net returns. We chose the beta distribution based on official statistics for crop farms from the MDB, reporting margins for various crops. Using data from the ABARE survey of MDB irrigation farms in 2006-2007, Hughes, Mackinnon, and Ashton (2009) estimated net returns per hectare for various crops and reported quartiles of their respective distributions. Estimated returns per hectare are defined as farm cash receipts less estimated unit costs and hence can be compared to the net benefit per unit of land used in our model. These estimates were used to calibrate the beta distributions of the net returns of irrigated wheat and dry-land wheat under the dry conditions scenario, since 2006-2007 was characterized by a severe drought and historical low water allocations. For the normal conditions scenario, we used estimations of gross margins reported by the New South Wales Department of Primary Industries for (irrigated) cotton and dry-land wheat; available at http://www.dpi.nsw.gov.au/content/agriculture/farm- business/budgets (accessed 25 March 2017).
Table 2: Average values of the main parameters, 50 draws Parameter Dry conditions Normal conditions
R¯ -181 1,936
¯
-223 351
¯
r 403 315
σR 351 1,056
σ 124 190
σr 174 209
σR 2,595 10,530
σrR -18,298 -60,685
σr -4,452 -7,526
hectare of the non-irrigated activity (ε) to be lower (due to a lower yield) when rainfall is scarce and hence when r is high. We thus also expect a negative correlation betweenε and r. We set the Spearman rank correlation at ρ2(ε, r) = −0.2. In order to generate pairs of random variables with some degree of dependence, we follow the procedure described in Johnson and Tenenbein (1981); see Appendix F for greater details.22
For both the normal and dry conditions scenarios, our simulation results are averages calculated over 50 draws of the following set of parameters: ¯R, ¯ε, ¯r,σε,σR,σr,σεR,σrR and σrε. Average values for these parameters under the dry and normal conditions are shown in Table 2.23
In order to simulate our theoretical model, values also need to be chosen for the following parameters:
• Initial farmer’s wealth: we set initial wealth (w0) at 5,000,000 AUD. This corresponds to the (farm average) total value of capital in fiscal year 2007-2008 for the irrigated broadacre farms in the MDB (Ashton, Hooper, and Oliver, 2010).
• Total farm land: we set total land (L) at 2,000 hectares. This corresponds to the average area operated on broadacre irrigated farms in the MDB in fiscal year 2007-2008 (Ashton and Oliver, 2011).
22This approach was used in Babcock and Hennessy (1996) to simulate price and crop yields.
23The results were almost unchanged when we increased the number of draws to 100 or 150.
• Initial water allocation: we set initial allocations (e0) at 2 megaliters (ML where 1ML equals 1 million liters) per hectare in the dry conditions scenario and at 3 ML per hectare in the normal conditions scenario. These figures were chosen based on the average water use for pastures and cereal crops in the MDB as reported by the Australian Bureau of Statistics.24
Under the above assumptions, the value of the initial water allocations (calculated using the average water price) represents 32% of the farmer’s initial wealth under dry conditions and 38% under normal conditions, which reflects the fact that the risk borne by farmers due to the uncertain water price is important relative to the non-random part of their wealth.
Finally we need to specify the mean-variance function. We chose a functional form for V such that all assumptions made in the theoretical section are satisfied: Vµ > 0, Vv < 0, Vµµ <0,Vvµ >0, αµ <0, and αv >0. Under these assumptions, farmers are characterized by DARA preferences and variance vulnerability. We assume:25
V(µ, v) =− 1 µ4−500v2. 6.2 Simulation results
We varied the level of initial water allocations (e0) from 0 to 5 ML/ha (in steps of 0.1 ML).
For each level ofe0 we searched for the proportion of land allocated to the irrigated crop (x) that maximized the mean-variance utility function. In each case we checked that the utility function satisfied the concavity condition.
In Figure 1a we show how the optimal land allocated to the irrigated crop (x∗) varied for different levels of the initial water allocation (e0) with and without forward trading under normal conditions. Similarly, Figure 1b shows how the optimal land allocation to the irrigated crop varied under dry conditions. We assumed unbiased expectations on the forward market.
24See http://www.abs.gov.au/ausstats/[email protected]/Lookup/4618.0main+features62011-12, accessed 20 April 2016.
25We tried a number of values for theγparameter and selected the one such that all the above theoretical assumptions about farmers’ preferences were satisfied and which permitted finding an optimal solution in terms of the land allocated to the irrigated crop.
Under normal conditions and without forward trading (dotted line in Figure 1a), the optimal share of irrigated land increased with the level of initial water allocations up to a levele0 =3.3 ML/ha and followed a very moderate decreasing trend for higher levels of water allocations.
The share of irrigated land was close to 65% when water allocations were at their maximum (e0 =5 ML/ha). Initial water allocations were found to be risk-increasing in 95% of the cases (see condition (4)) so the impact of the initial endowment on the share of irrigated land depended on the relative magnitude of the correlation effect (CE), mean effect (ME) and variance effect (VE) as described in Proposition 2. As shown in Figure 2a, the variance effect (VE) outweighed the correlation effect (CE) and the mean effect (ME) for values above e0 =3.3, which explains the (moderate) decreasing trend in the share of irrigated land for initial water allocations greater than 3.3 ML/ha.
Under dry conditions and without forward trading (dotted line in Figure 1b), the share of irrigated land increased for water allocations up to e0 =0.7 ML/ha, decreased for values ofe0 above 0.8 ML/ha and below 1.6 ML/ha, and followed an upward trend for values above e0 =1.6 ML/ha. Similarly to the normal conditions scenario, the initial endowment was risk-increasing in almost all cases (94%) so the (positive or negative) impact of the initial water allocations on the share of irrigated land was driven by the relative magnitude of the correlation, mean, and variance effects. As shown in Figure 2b, the variance effect outweighed the mean and correlation effects for values of e0 = above 0.8 ML/ha and below 1.6 ML/ha.
Above 1.6 ML/ha, the correlation effect followed a sharply increasing trend: the increase in the initial water endowment increased the background risk and induced the farmer to allocate more land to the irrigated crop because it was more strongly negatively correlated with the background risk than the rain-fed crop.
The possibility of forward trading induced the farmer (whose anticipations are unbiased) to allocate more land to the irrigated crop when the level of initial water allocations increased under both normal and dry conditions, which confirms Proposition 4. In most cases, the possibility of forward trading increased the share of irrigated land compared to the situation
